Processes in the solid Earth are inaccessible to direct observation. However, the effect of mass redistribution and deformation of the solid Earth can be recorded using satellite observations and observations on the Earth's surface. The measurement results usually contain a superposition of many processes. Numerical models are a useful tool for simplifying and separating processes of the solid Earth. This simplified representation of nature contains a mechanical description of the solid Earth.
The solid Earth extends from its surface to its core and generally does not include the oceans, atmosphere or ice sheets. The mechanical properties of the solid Earth are described using a continuum mechanics approach. This means that the Earth is split into small volume elements for which the corresponding material properties are described. In a good approximation, the material properties such as density or elasticity only change with depth, which leads to a spherically symmetrical structure. General physical principles are used to describe the interaction of the volume elements. Integration over the volume of the earth body then defines its response to an acting force. For the surface and tidal forces considered in this section, the lithosphere and the Earth's mantle are of main interest. Depending on the time scale on which the processes take place, elastic or viscous material behavior can be assumed.
References:
Klemann, V. (2024 online): Gravito-viscoelastodynamics. - In: Sideris, M. G. (Ed.), Encyclopedia of Geodesy, (Encyclopedia of Earth Sciences Series (EESS)), Cham : Springer International Publishing, 1-6; doi.org/10.1007/978-3-319-02370-0_141-1
The elastic response to surface forces can be adequately parameterized by linear transfer functions, Green's functions, which are based on a radially symmetrical density and elasticity distribution in the earth's interior. The respective earth structure has to be adapted for small-scale processes.
For the investigation of surface deformations caused by seasonal changes in the terrestrial water cycle or in the atmosphere, the deformation of the earth's surface was calculated as a function of the respective elastic crust (earth structure). For this purpose, a separate set of Green's functions was calculated as a first approximation at each grid point for a laterally varying crustal structure. These describe the reaction of the solid earth as a function of the distance to the load change.
It is numerically more expensive to consistently model lateral variations in the elastic properties. These occur mainly at the tectonic plate boundaries and influence the deformation behavior. Density variations, although also relevant, have less influence. In the Earth's mantle, there is another phenomenon, anelasticity, which means that the elastic moduli decrease with the duration of the loading and the material deforms to a larger degree. Similar to the crust, also in the mantle the anelastic properties vary from place to place. They strongly depend on temperature, so their quanitfication is closely linked to studies on the dynamics of the Earth's mantle. (See also Department 2 "Geophysics")
References:
Tanaka, Y., Klemann, V., Martinec, Z. (2024): An estimate of the effect of 3D heterogeneous density distribution on coseismic deformation using a spectral finite-element approach. - In: Freymueller, J.T., Sánchez, L. (eds) X Hotine-Marussi Symposium on Mathematical Geodesy. HMS 2022. International Association of Geodesy Symposia, vol 155. Springer, Cham. p 103-111, doi.org/10.1007/1345_2023_236
Huang, P., Sulzbach, R., Klemann, V., Tanaka, Y., Dobslaw, H., Martinec, Z., Thomas, M. (2022): The influence of sediments, lithosphere and upper mantle (anelastic) with lateral heterogeneity on ocean tide loading and ocean tide dynamics. - Journal of Geophysical Research: Solid Earth, 127, 11, e2022JB025200. doi.org/10.1029/2022JB025200
Huang, P., Sulzbach, R., Tanaka, Y., Klemann, V., Dobslaw, H., Martinec, Z., Thomas, M. (2021): Anelasticity and lateral heterogeneities in Earth's upper mantle: impact on surface displacements, self-attraction and loading and ocean tide dynamics. - Journal of Geophysical Research: Solid Earth, 126, 9, e2021JB022332. doi.org/10.1029/2021JB022332
Tanaka, Y., Klemann, V., Martinec, Z. (2019): Surface Loading of a Self-Gravitating, Laterally Heterogeneous Elastic Sphere: Preliminary Result for the 2D Case. - In: IX Hotine-Marussi Symposium on Mathematical Geodesy, (International Association of Geodesy Symposia ; 151), 157-163. https://doi.org/10.1007/1345_2019_62
Dill, R., Klemann, V., Martinec, Z., Tesauro, M. (2015): Applying local Green's functions to study the influence of the crustal structure on hydrological loading displacements. - Journal of Geodynamics, 88, p. 14-22.| doi.org/10.1016/j.jog.2015.04.005
For processes such as the glacial isostatic adjustment GIA, the viscous behavior of the earth's mantle must be taken into account in addition to the elastic behavior. The viscoelastic behavior caused that during the last glacial maximum, around 20,000 years ago, the ground in the glaciated areas sank by up to one kilometer and the ground is still rising after the ice sheets have melted
See also: Glacial isostatic adjustment
References:
Bagge, M., Klemann, V., Steinberger, B., Latinovic, M., Thomas, M. (2021): Glacial-isostatic adjustment models using geodynamically constrained 3D Earth structures. - Geochemistry Geophysics Geosystems (G3), 22, 11, e2021GC009853. doi.org/10.1029/2021GC009853
Tanaka, Y., Hasegawa, T., Tsuruoka, H., Klemann, V., Martinec, Z. (2015): Spectral-finite element approach to post-seismic relaxation in a spherical compressible Earth: application to gravity changes due to the 2004 Sumatra-Andaman earthquake. - Geophysical Journal International, 200, p. 299-321. doi.org/10.1093/gji/ggu391
Cambiotti, G., Klemann, V., Sabadini, R. (2013): Compressible viscoelastodynamics of a spherical body at long timescales and its isostatic equilibrium. - Geophysical Journal International, 193, 3, p. 1071-1082. doi.org/10.1093/gji/ggt026
Spada, G., Barletta, V. R., Klemann, V., Riva, R. E. M., Martinec, Z., Gasperini, P., Lund, B., Wolf, D., Vermeersen, L. L. A., King, M. (2011): A benchmark study for glacial isostatic adjustment codes. - Geophysical Journal International, 185, 106-132; doi.org/10.1111/j.1365-246X.2011.04952.x
Klemann, V., Martinec, Z., Ivins, E. R. (2008): Glacial isostasy and plate motion. - Journal of Geodynamics, 46, 3-5, p. 95-103. doi.org/10.1016/j.jog.2008.04.005